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8.1 Multiplication 8.1 Multiplication Properties of ExponentsProperties of Exponents
Multiplying Monomials and Raising Monomials to Powers
Objective: Use properties of exponents to multiply exponential expressions.
Vocabulary
• Monomials - a number, a variable, or a product of a number and one or more variables
• 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials.
• Constant – a monomial that is a number without a variable.
• Base – In an expression of the form xn, the base is x.
• Exponent – In an expression of the form xn, the exponent is n.
Writing - Using ExponentsWriting - Using Exponents
Rewrite the following expressions using exponents:
x x x x y y The variables, x and y, represent the bases. The number of times each base is multiplied by itself will be the value of the exponent.
x x x x y y x y 4 2
Writing Expressions without Exponents
Write out each expression without exponents (as multiplication):
8a3b2 8a a a b b
xy 4 xyxyxyxy
or
xxxxy yyy
Simplify the following expression: (5a2)(a5)
Step 1: Write out the expressions in expanded form.
Step 2: Rewrite using exponents.
Product of PowersProduct of Powers
5a2 a5 5a a aa a a a
There are two monomials. Underline them.
What operation is between the two monomials?
Multiplication!
5a2 a5 5a7 5a7
For any nonzero number a, and all integers m and n,
am • an = am+n.
Product of Powers RuleProduct of Powers Rule
1) a9 a4 a13
2) w2 w10 w12
4) k5 k3 5) x 2 y2
3) r r5 r 6
x2 y2
k8
Notice that the Produce of Power Rule is only for those powers of the SAME BASE
If the monomials have coefficients, multiply those, but still add the powers.
Multiplying MonomialsMultiplying Monomials
1) 4a9 2a4 8a13
2) 7w2 10w10 70w12
4) 3k 5 7k 3 5) 12x2 2y2
3) 2r 3r5 6r 6
24x2 y2
21k8
These monomials have a mixture of different variables. Only add powers of like variables.
Multiplying MonomialsMultiplying Monomials
1) 4a9b3 2a4b 8a13b4
2) 7w2 y5 10w10y2 70w12y7
4) 3k 5mn4 7k3m3n3 5) 12x2 y3 2xy2
3) 2rt3 3r5 6r 6t 3
24x3 y5
21k8m4n7
Simplify the following: ( x3 ) 4
Note: 3 x 4 = 12.
Power of PowersPower of Powers
The monomial is the term inside the parentheses.
Step 1: Write out the expression in expanded form.
x3 4x3 x3 x3 x3
xxx x x x x x xxxx
x3 4x12
Step 2: Simplify, writing as a power.
Power of Powers RulePower of Powers Rule
For any nonzero number, a, and all integers m and n,
am namn.
1) b9 10 b90
2) c3 3 c9
3) w12 2 w 24
Monomials to PowersMonomials to Powers
1) 2b9 3 8b27
2) 5c3 3 125c9
3) 7w12 2 49w24
If the monomial inside the parentheses has a coefficient, raise the coefficient to the power, but still multiply the variable powers.
Monomials to Powers(Power of a Product)Monomials to Powers(Power of a Product)
5w3xy2 4 5 4
w3 4x 4
y2 4
625w34 x4 y24
If the monomial inside the parentheses has more than one variable, raise each variable to the outside power using the Power of a Product rule.
(ab)m = am•bm
625w12x4 y8
Monomials to Powers(Power of a Product)Monomials to Powers(Power of a Product)
1) 2b9c4 3 8b27c12
2) 5a5c3 3125a15c9
3) 7w12y4z 2 49w24y8z2
Simplify each expression:
SummarySummaryRemember the properties of exponents.1. Product of Power Rule
For any nonzero number a, and all integers m and n,
2. Power of Power Rule
For any nonzero number, a, and all integers m and n,
nm mna a
nmnm aaa
3. Power of Product Rule
For any nonzero numbers, a and b, and integer m,
mmm baab
Guided Practice – L 8.1 DHQ
PAGE: 453 in Textbook
#’s 4-20 EVEN ONLY
Assignment – Lesson 8.1Assignment – Lesson 8.1
P. 453
#’s 22-31, 34-45, 52-60
Chapter 8Exponents
Section 8.2
Zero and Negative Exponents
45 + 3
48
y3 + 4 + 5
y12
21 + 6
27
(-5)1 + 3
(-5)4
54
x3(2)
x6
52(3)
56
(-2)3(4)
(-2)12
212
(a – 2)3(2)
(a – 2)6
34 x4 y4 (-3)2 y2
9y2
-32y2
-9y2
32 x8 y2 • y5
9x8y7
37 z7 56
48 y15 -415
25 x16 y5
431
1y4
1
1
1
53
35 Undefined
k31
4
k34
43 h1
g1
5
43hg5
4-3 + 3
40
1
5(-2)(-3)
5632
1
81
3)y4(1
33 y41
3y641 g2n
3
2h3 2
4
k m
60 = 13
1
(4 2)n
SummaryRemember the properties of zero and negative exponents
4. Zero Exponent Rule
For any nonzero number a,
5. Negative Exponent Rule
For any nonzero number, a, and all integer n,
1nn
aa
0 1a
1 nn
aa
Guided Practice – L 8.2 DHQ
PAGE: 459 in Textbook
#’s 3-11 ODD ONLY
Assignment – Lesson 8.2Assignment – Lesson 8.2
P. 459-460
#’s 14-44 EVEN ONLY + #’s 46-48 ALL
Chapter 8ExponentsDivision (Quotient) Properties of Exponents
Notice that when answering the questions, you have to combine all the rules you have learned, not just the rules you learned in this section.
= 15 44 4
8
9
6
6 8 96 16
1
6
2
4
2x
x 2 42x 22x 2
2
x
=103 1
3 6y
= a51
64 2
1
c
Power of Product, Power of Power
3
3
1
4
1
64
2
2
4
x
2
16
x
4
4
3
2
4
4
2
3
54x
z
20
5
x
z
5 3
3
36
4
x y
xy 5 1 3 39x y 4 09x y 49x
3 6
3
3 x
y
6
3
27x
y
a. = (x5 – 3)-1 = (x2)-1 = x-22
1
x
3 6 3
2 2 6
2
2
a b
a b 3 2 6 2 3 62 a b 4 32a b
4
3
2a
b
Remember! Simplify the inside expression first!!!
1
71
4
4
2
m
27
125
-4m4n2
2
4
9y
r
3. 4. 5.
6. 7.
3
1
x
1
m 1
4 22
3
z
3 327
8
m n
Summary
1. The division property of the exponents is only for the power of same base.
2. The other learned properties may be used at the same time.
SummaryRemember the properties of Quotient of Power and Power of Quotient.
6. Quotient of Power Rule
For any nonzero number a, and all integers m and n,
7. Power of Quotient Rule
For any nonzero number, a, b, and all integer m,
, 0, 0m m
m
a aa b
b b
, 0m
m nn
aa a
a
Guided Practice – L 8.3 DHQ
PAGE: 466 in Textbook
#’s 3-5, 11-13
Assignment – Lesson 8.3Assignment – Lesson 8.3
P. 466
#’s 19-48 ALL
Objectives
1. Write and use the models for exponential growth.
2. Write and use the models for exponential decay.
Exponential Growth Functions8.5
EXPONENTIAL GROWTH MODEL
A quantity is growing exponentially if it increases by the same percent in each time period.
C is the initial amount. t is the time period.
(1 + r) is the growth factor, r is the growth rate.
The percent of increase is 100r.
y = C (1 + r)t
Sometimes use P instead of C Note: measure of rate and time MUST be in the same time unit
Example 1 Compound Interest
You deposit $1500 in an account that pays 2.3% interest compounded yearly,
1)What was the initial principal (P) invested?
2)What is the growth rate (r)? The growth factor?
3)Using the equation A = P(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money?
C or P = $1500
growth rate (r) is 0.023. The growth factor is 1.023
y = $1569.79
1.What is the percent increase each year?
2.Write a model for the number of rabbits in any given year.
3.Find the number of rabbits after 5 years.
Example 2 Exponential Growth Model
A population of 20 rabbits is released into a wild-life region. The population triples each year for 5 years.
200%
≈ 4860 rabbits
R =20(1+2.0
0)t
Example 2 Exponential Growth Model
Graph the growth of the rabbits.
Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points.
t
P 486060 180 540 162020
51 2 3 40
0
1000
2000
3000
4000
5000
6000
1 72 3 4 5 6Time (years)
Po
pu
lati
on
P = 20 ( 3 ) t Here, the large
growth factor of 3 corresponds to a rapid increase
Here, the large growth factor of 3 corresponds to a rapid increase
EXPONENTIAL DECAY MODEL
C is the initial amount.t is the time period.
(1 – r ) is the decay factor, r is the decay rate.
The percent of decrease is 100r.
y = C (1 – r)t
Exponential Decay Functions8.6A quantity is decreasing exponentially if it decreases by the same percent in each time period.
Sometimes use P instead of C Note: measure of rate and time MUST be in the same time unit
Example 4 Asset Depreciation
You buy a new car at $22,500. The car depreciates at the rate of 7% per year.
1.What was the initial amount invested?
2.What is the decay rate? The decay factor?
3.What will the car be worth after the first year? The second year?
C or P = $22,500
growth rate (r) is 0.07. The growth factor is .93
y1 = $20,925 y2 = $19,460.25
EXPONENTIAL GROWTH AND DECAY MODELS
y = C (1 – r)ty = C (1 + r)t
EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL
1 + r > 1 0 < 1 – r < 1
CONCEPT
SUMMARY
An exponential model y = a • b t represents exponential
growth if b > 1 and exponential decay if 0 < b < 1.C is the initial amount.t is the time period.
(1 – r) is the decay factor, r is the decay rate.
(1 + r) is the growth factor, r is the growth rate. (0, C)(0, C)
Example 6 Exponential Growth or Decay
Your business had a profit of $25,000 in 1998. If the profit increased by 12% each year, what would your expected profit be in the year 2010? Identify C, t, r, and the growth factor. Write down the equation you would use and solve.
y = $97,399.40
Example 6 Exponential Growth or Decay
Iodine-131 is a radioactive isotope used in medicine. Its decay rate is 50%. If a patient is given 25mg of iodine-131, how much would be left after 32 days or 4 half-lives. Identify C, t, r, and the decay factor. Write down the equation you would use and solve.
y = 1.56 mg
The graph of exponential growth function is
C
The graph of exponential growth function is like an air plane taking off.
The graph of exponential decay function is
C
The graph of exponential decay function is like an air plane landing.
GRAPHING EXPONENTIAL DECAY MODELS
EXPONENTIAL GROWTH AND DECAY MODELS
y = C (1 – r)ty = C (1 + r)t
EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL
1 + r > 1 0 < 1 – r < 1
CONCEPT
SUMMARY
An exponential model y = a • b t represents exponential
growth if b > 1 and exponential decay if 0 < b < 1.C is the initial amount.t is the time period.
(1 – r) is the decay factor, r is the decay rate.
(1 + r) is the growth factor, r is the growth rate. (0, C)(0, C)
1. Exponential growth and decay models arey = C(1 + r)t y =
C(1 – r)t
Or they can be combined to one:y = C(1 ± r)t
2. Their graphs are always pass through point (0, C).
3. The graph of exponential growth functions is strictly increasing. The graph of exponential decay functions is strictly decreasing. Their graphs always stay above the x-axis.
Guided Practice – L 8.5/8.6 DHQ
PAGE: 480 in Textbook
# 4
PAGE: 488 in Textbook
# 5
PLEASE SHOW WORK!
Assignment – Lesson 8.5 + 8.6
Assignment – Lesson 8.5 + 8.6
P. 480 #’s 6-11
P. 488 #’s 10-16
SHOW WORK!
